Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C2×#99 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×Q8 | C23×C17 |
| prime radical | C2×#41×C17 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C23×C17 ◁ C22×Q8×C17 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,26,23,27,38,45)(20,28,31,46,29,42,36,48)(21,32,34,44,35,25,37,43)(24,39,41,33,40,47,49,30);(18,20)(19,24)(21,33)(22,36)(23,29)(25,42)(26,41)(27,40)(28,32)(30,35)(31,38)(34,39)(37,47)(43,48)(44,46)(45,49);(20,29)(24,40)(28,42)(30,33)(31,36)(39,47)(41,49)(46,48)391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 865);